Expression of type Equals

from the theory of proveit.physics.quantum.QPE

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import k, t
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = Add(t, one)
sub_expr3 = Add(k, two_pow_t)
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, sub_expr2)), domain = Interval(two_pow_t, subtract(Mult(two, two_pow_t), one))), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, sub_expr3)), NumKet(sub_expr3, sub_expr2)), domain = Interval(zero, subtract(two_pow_t, one))))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(\sum_{k=2^{t}}^{\left(2 \cdot 2^{t}\right) - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t + 1}\right)\right) = \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot \left(k + 2^{t}\right)} \cdot \lvert k + 2^{t} \rangle_{t + 1}\right)\right)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameter: 61 body: 10
9	Lambda	parameter: 61 body: 12
10	Conditional	value: 13 condition: 14
11	ExprTuple	61
12	Conditional	value: 15 condition: 16
13	Operation	operator: 19 operands: 17
14	Operation	operator: 21 operands: 18
15	Operation	operator: 19 operands: 20
16	Operation	operator: 21 operands: 22
17	ExprTuple	23, 24
18	ExprTuple	61, 25
19	Literal
20	ExprTuple	26, 27
21	Literal
22	ExprTuple	61, 28
23	Operation	operator: 64 operands: 29
24	Operation	operator: 33 operands: 30
25	Operation	operator: 35 operands: 31
26	Operation	operator: 64 operands: 32
27	Operation	operator: 33 operands: 34
28	Operation	operator: 35 operands: 36
29	ExprTuple	39, 37
30	ExprTuple	61, 41
31	ExprTuple	62, 38
32	ExprTuple	39, 40
33	Literal
34	ExprTuple	53, 41
35	Literal
36	ExprTuple	42, 43
37	Operation	operator: 55 operands: 44
38	Operation	operator: 57 operands: 45
39	Literal
40	Operation	operator: 55 operands: 46
41	Operation	operator: 57 operands: 47
42	Literal
43	Operation	operator: 57 operands: 48
44	ExprTuple	66, 50, 51, 52, 61
45	ExprTuple	49, 54
46	ExprTuple	66, 50, 51, 52, 53
47	ExprTuple	67, 63
48	ExprTuple	62, 54
49	Operation	operator: 55 operands: 56
50	Literal
51	Literal
52	Literal
53	Operation	operator: 57 operands: 58
54	Operation	operator: 59 operand: 63
55	Literal
56	ExprTuple	66, 62
57	Literal
58	ExprTuple	61, 62
59	Literal
60	ExprTuple	63
61	Variable
62	Operation	operator: 64 operands: 65
63	Literal
64	Literal
65	ExprTuple	66, 67
66	Literal
67	Variable